How come ambient occlusion shadows not showing in Blender Render-Render?
The viewport AO feature is fairly new, and has extra options that allow it to be overdriven and produce extra high ao for the the edges between the cube and the plane. An open 90 degree corner like that should not really produce much ambient occlusion because it's too open for much occlusion to happen. The older AO feature in the World panel is actually working more accurately than the viewport tool.

You can actually drive the AO value high by manually typing in a value higher than 1.0.

In my example I've added an inset edge in the middle of the cube. You can now see more clearly the difference between the Env light only shadow and the addition of the AO shadow. Note that the surface does not get brigher with the AO set to Multiply.